We study the well-posedness of second order Hamilton-Jacobi equations with an Ornstein-Uhlenbeck operator in $\mathbb{R}^N$ and $\mathbb{R}^N× [0, +∞).$ As applications, we solve the associated ergodic problem associated to the stationary equation and obtain the large time behavior of the solutions of the evolution equation when it is nondegenerate. These results are some generalizations of the ones obtained by Fujita, Ishii & Loreti 2006 [19] by considering more general diffusion matrices or nonlocal operators of integro-differential type and general sublinear Hamiltonians. Our work uses as a key ingredient the a-priori Lipschitz estimates obtained in Chasseigne, Ley & Nguyen 2017 [10].